Variogram Estimation Research Articles

Spatial design matrices and associated quadratic forms: structure and properties

The paper provides significant simplifications and extensions of results obtained by Gorsich, Genton, and Strang (J. Multivariate Anal. 80 (2002) 138) on the structure of spatial design matrices. These are the matrices implicitly defined by quadratic forms that arise naturally in modelling intrinsically stationary and isotropic spatial processes. We give concise structural formulae for these matrices, and simple generating functions for them. The generating functions provide formulae for the cumulants of the quadratic forms of interest when the process is Gaussian, second-order stationary and isotropic. We use these to study the statistical properties of the associated quadratic forms, in particular those of the classical variogram estimator, under several assumptions about the actual variogram.

Estimating Variogram Uncertainty

The variogram is central to any geostatistical survey, but the precision of a variogram estimated from sample data by the method of moments is unknown. It is important to be able to quantify variogram uncertainty to ensure that the variogram estimate is sufficiently accurate for kriging. In previous studies theoretical expressions have been derived to approximate uncertainty in both estimates of the experimental variogram and fitted variogram models. These expressions rely upon various statistical assumptions about the data and are largely untested. They express variogram uncertainty as functions of the sampling positions and the underlying variogram. Thus the expressions can be used to design efficient sampling schemes for estimating a particular variogram. Extensive simulation tests show that for a Gaussian variable with a known variogram, the expression for the uncertainty of the experimental variogram estimate is accurate. In practice however, the variogram of the variable is unknown and the fitted variogram model must be used instead. For sampling schemes of 100 points or more this has only a small effect on the accuracy of the uncertainty estimate. The theoretical expressions for the uncertainty of fitted variogram models generally overestimate the precision of fitted parameters. The uncertainty of the fitted parameters can be determined more accurately by simulating multiple experimental variograms and fitting variogram models to these. The tests emphasize the importance of distinguishing between the variogram of the field being surveyed and the variogram of the random process which generated the field. These variograms are not necessarily identical. Most studies of variogram uncertainty describe the uncertainty associated with the variogram of the random process. Generally however, it is the variogram of the field being surveyed which is of interest. For intensive sampling schemes, estimates of the field variogram are significantly more precise than estimates of the random process variogram. It is important, when designing efficient sampling schemes or fitting variogram models, that the appropriate expression for variogram uncertainty is applied.

Perspectives on mapping the MARGO reconstructions by variogram analysis/kriging and objective analysis

Paleo-data are not always useful in their original scattered distribution: For many numerical modeling issues and for display and comparison, gridded versions that provide meaningful estimates for under-sampled regions are a must. We constructed a test data set with a spatial resolution identical to the Multi-proxy Approach for the Reconstruction of the Glacial Ocean (MARGO) samples from the World Ocean Atlas temperatures and assess the performance of (i) variogram estimation and kriging and (ii) the Levitus objective analysis in reconstructing the original data. The two methods complement each other with respect to the facility of application and the quality of the results. Kriging requires a careful parameter adjustment but delivers the smallest deviation from the original data (1.22 °C in the global average), whereas the Levitus analysis provides a fast and efficient tool for checking the samples from different proxy data against each other during the compilation of the final MARGO database, at the expense of a slightly higher error (1.56 °C).

Spatial statistics of clustered data

Modern spatial statistics techniques are widely used to make predictions for natural processes that are continuously distributed over some convex domain. Implementation of these techniques often relies on the adequate estimation of certain spatial correlation functions such as the covariance and the variogram from the data sets available. This work studies the practical estimation of such spatial correlation functions in the case of clustered data. The coefficient of variation of the dimensionless spatial density of the point pattern of sample locations is suggested as a useful metric for degree of clusteredness of the clustered data set. We show that the common variogram estimator becomes increasingly unreliable with increasing coefficient of variation of the dimensionless spatial density of the point pattern of sample locations. Moreover, we present a modified form of the variogram estimator that incorporates declustering weights, and propose a scheme for estimating the declustering weights based on zones of proximity. Finally, insight is gained in terms of a numerical application of the common and modified methods on piezometric head data collected over an irregular network.

Moments of random vectors with skew t distribution and their quadratic forms

Moments of skew t random vectors and their quadratic forms are derived. It is shown that the moments of the sample autocovariance function and of the sample variogram estimator depend on a measure of multivariate kurtosis, but not on a skewness parameter.

Least Squares Variogram Fitting by Spatial Subsampling

SummaryLeast squares methods are popular for fitting valid variogram models to spatial data. The paper proposes a new least squares method based on spatial subsampling for variogram model fitting. We show that the method proposed is statistically efficient among a class of least squares methods, including the generalized least squares method. Further, it is computationally much simpler than the generalized least squares method. The method produces valid variogram estimators under very mild regularity conditions on the underlying random field and may be applied with different choices of the generic variogram estimator without analytical calculation. An extension of the method proposed to a class of spatial regression models is illustrated with a real data example. Results from a simulation study on finite sample properties of the method are also reported.

Nonparametric variogram and covariogram estimation with Fourier–Bessel matrices

The nonparametric estimation of variograms and covariograms for isotropic stationary spatial stochastic processes is considered. It is shown that Fourier–Bessel matrices play an important role in this context because they provide an orthogonal discretization of the spectral representation of positive definite functions. Their properties are investigated and an example from a simulated two-dimensional spatial process is provided. It is shown that this approach provides a smooth and positive definite nonparametric estimator in the continuum, whereas previous methods typically suffer from spurious oscillations. A practical example from Astronomy is used for illustration.

Eigenstructures of Spatial Design Matrices

In estimating the variogram of a spatial stochastic process, we use a spatial design matrix. This matrix is the key to Matheron's variogram estimator. We show how the structure of the matrix for any dimension is based on the one-dimensional spatial design matrix, and we compute explicit eigenvalues and eigenvectors for all dimensions. This design matrix involves Kronecker products of second order finite difference matrices, with cosine eigenvectors and eigenvalues. Using the eigenvalues of the spatial design matrix, the statistics of Matheron's variogram estimator are determined. Finally, a small simulation study is performed.

A statistical evaluation of non-ergodic variogram estimators

Geostatistics is a set of statistical techniques that is increasingly used to characterize spatial dependence in spatially referenced ecological data. A common feature of geostatistics is predicting values at unsampled locations from nearby samples using the kriging algorithm. Modeling spatial dependence in sampled data is necessary before kriging and is usually accomplished with the variogram and its traditional estimator. Other types of estimators, known as non-ergodic estimators, have been used in ecological applications. Non-ergodic estimators were originally suggested as a method of choice when sampled data are preferentially located and exhibit a skewed frequency distribution. Preferentially located samples can occur, for example, when areas with high values are sampled more intensely than other areas. In earlier studies the visual appearance of variograms from traditional and non-ergodic estimators were compared. Here we evaluate the estimators' relative performance in prediction. We also show algebraically that a non-ergodic version of the variogram is equivalent to the traditional variogram estimator. Simulations, designed to investigate the effects of data skewness and preferential sampling on variogram estimation and kriging, showed the traditional variogram estimator outperforms the non-ergodic estimators under these conditions. We also analyzed data on carabid beetle abundance, which exhibited large-scale spatial variability (trend) and a skewed frequency distribution. Detrending data followed by robust estimation of the residual variogram is demonstrated to be a successful alternative to the non-ergodic approach.

The change-of-variance function: a tool to explore the effects of dependencies in spatial statistics

Abstract This paper presents the computation of the change-of-variance function of M-estimators of scale under general contamination for dependent observations. In this context, several results of robustness are established, and the links between B -robustness, V -robustness and V ♢ -robustness are studied. Some more specific properties are derived for Gaussian distributions. These results are then applied to variogram estimation, which is a crucial stage of spatial prediction. The change-of-variance function is shown to be a tool to explore the effects of dependencies on the variance of variogram estimators. ARMA models are used in order to model unidirectional spatial dependencies. It is shown that the shape of the change-of-variance function under dependence is characteristic of the type of variogram estimator. However, this shape depends also on the underlying dependency structure, its intensity, as well as the lag distance of the variogram estimates. Therefore, statistical insight is provided into the sensitivity and the behavior of the variance of the variogram estimator at different spatial lags. For instance, this variance plays an important role when fitting a parametric variogram model by weighted or generalized least squares.

Moments of skew-normal random vectors and their quadratic forms

In this paper, we derive the moments of random vectors with multivariate skew-normal distribution and their quadratic forms. Applications to time series and spatial statistics are discussed. In particular, it is shown that the moments of the sample autocovariance function and of the sample variogram estimator do not depend on the skewness vector.

Variance–Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty

Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach.

Computing geostatistical image texture for remotely sensed data classification

Most classical mathematical algorithms for image classification do not usually consider the spectral dependence existing between a pixel and its neighbours, i.e., spatial autocorrelation. Thus, it would be advisable for discrimination of landcover classes to add to the radiometric bands of the sensor complementary information related to the textural features of an image, which can be analysed from the autocorrelation spatial structure of the digital numbers. In this way, the results obtained from pixel-by-pixel classifiers simultaneously taking into account both radiometric and texture information could be improved. This improvement would arise from the hypothesis that a pixel is not independent of its neighbours and, furthermore, that its dependence can be quantified and incorporated into the classifier. In this paper we present a methodology based on computing a set of univariate and multivariate textural measures of spatial variability based on several variogram estimators. Madogram and direct variogram for the univariate case, and cross and pseudo-cross variograms for the multivariate one, have been proposed. These measures are calculated for a specific lag of distance in a neighbourhood using a moving window on the two most representative principal components of the radiometric bands, enabling us to quantify the spatial variability of radiometric data at a local level. A computer program has been written to create a multiband image texture as output file that can be used within the classification process as additional information. An application of this methodology to lithological discrimination is presented using a Landsat-5 TM image.

The correlation structure of matheron's classical variogram estimator under elliptically contoured distributions

The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed.

Robust principal component and factor analysis in the geostatistical treatment of environmental data

In this paper we show the usage of robust multivariate statistical methods in geostatistics. A usual procedure to estimate the values of variables (e.g. geochemical variables) measured at certain points of a region is to apply geostatistical methods like Krige estimation (based on the estimation of variograms). Here we emphasize robust principal component and factor analysis for the preliminary investigation of the data to reduce the dimension. Geostatistical methods are applied afterwards to the estimated factor scores. The final results show the influence of certain combinations of variables in the considered region. Moreover, the estimated factor scores with the robust procedure indicate outlying observations in a much better way. Copyright © 1999 John Wiley & Sons, Ltd.

Optimal spatial sampling design for the estimation of the variogram based on a least squares approach

A methodology for the choice of optimal spatial sampling designs is proposed. The problem is to find an optimal finite set of space locations where a random field has to be sampled, in order to minimize the variability of the parametric variogram estimator. Under the hypothesis that the random field is Gaussian second‐order stationary and using a first‐order approximation for the linearization of the parametric variogram with respect to its parameters, we are able to express the covariance matrix of the parameter estimators as a function of the sampling design. An optimization algorithm based on a generalized least squares approach is proposed in order to reduce the variability of these estimators through the minimization of the determinant of their covariance matrix. In order to validate this approach, a practical case study is conducted for two different variogram models. It shows that compared to random sampling designs, the benefit of the optimization procedure is somewhat limited for a variogram model without nugget effect. For a variogram model with nugget effect, random sampling designs are associated with much higher variability on the parameter estimators than optimized designs because of the typical lack of information offered by random sampling designs for small distances between locations. The performance of random sampling designs compared to optimized or alternative regular designs is shown to be poor with respect to parameter estimation, especially when a nugget effect is included in the variogram model. The parameters that benefit the most from the optimization procedure are the variance and the nugget effect, whereas the improvement for the range parameter estimation is limited. The optimization algorithm provides yardstick results, yielding reference values for the selection of an alternative regular design. The applicability of the algorithm is very wide, and it can greatly help the user to understand the way the parameters are influenced by the choice of a set of sampling locations instead of another.

Least-squares fitting from the variogram cloud

Several authors have noticed poor finite sample behavior of the most widely used parametric variogram estimator by Cressie (1985). This behavior is most notable in the case that fitting is done on the collection of squared differences of observations from all point pairs, the so-called variogram cloud. This can indeed be made plausible by asymptotical considerations and these motivate a corresponding simple correction. An alternative, even more efficient computer-intensive estimator is also proposed. The benefits of both methods are demonstrated by Monte-Carlo simulation.

Constrained optimisation of soil sampling for minimisation of the kriging variance

This paper introduces the extended Spatial Simulated Annealing (SSA) method to optimise spatial sampling schemes for obtaining the minimal kriging variance. Sampling schemes are optimised at the point level. Boundaries and previous observations can be taken into account. This procedure extends ordinary SSA which focuses entirely on variogram estimation and even distribution of observations over the area. We applied it to texture and phosphate content on a river terrace in Thailand. First, sampling was conducted for estimation of the variogram using ordinary SSA. The variograms thus obtained were used in extended SSA, yielding a reduction of the mean kriging variance of the sand percentage from 28.2 to 23.7(%) 2. The variograms were used subsequently in a geomorphologically similar area. Optimised sampling schemes for anisotropic variables differed considerably from those for isotropic ones. The schemes were especially efficient in reducing high values of the kriging variance near boundaries of the area.

Constrained Optimization of Spatial Sampling using Continuous Simulated Annealing

AbstractSpatial sampling is an important issue in environmental studies because the sample configuration influences both costs and effectiveness of a survey. Practical sampling constraints and available preinformation can help to optimize the sampling scheme. In this paper, spatial simulated annealing (SSA) is presented as a method to optimize spatial environmental sampling schemes. Sampling schemes are optimized at the point‐level, taking into account sampling constraints and preliminary observations. Two optimization criteria have been used. The first optimizes even spreading of the points over a region, whereas the second optimizes variogram estimation using a proposed criterion from the literature. For several examples it is shown that SSA is superior to conventional methods of designing sampling schemes. Improvements up to 30% occur for the first criterion, and an almost complete solution is found for the second criterion. Spatial simulated annealing is especially useful in studies with many sampling constraints. It is flexible in implementing additional, quantitative criteria.

Spatial Breakdown Point of Variogram Estimators

In the context of robust statistics, the breakdown point of an estimator is an important feature of reliability. It measures the highest fraction of contamination in the data that an estimator can support before being destroyed. In geostatistics, variogram estimators are based on measurements taken at various spatial locations. The classical notion of breakdown point needs to be extended to a spatial one, depending on the construction of most unfavorable configurations of perturbation. Explicit upper and lower bounds are available for the spatial breakdown point in the regular unidimensional case. The difficulties arising in the multidimensional case are presented on an easy example in IR2, as well as some simulations on irregular grids. In order to study the global effects of perturbations on variogram estimators, further simulations are carried out on data located on a regular or irregular bidimensional grid. Results show that if variogram estimation is performed with a 50% classical breakdown point scale estimator, the number of initial data likely to be contaminated before destruction of the estimator is roughly 30% on average. Theoretical results confirm the previous statement on data in IRd, d ≥ 1.